Mixed numbers and improper fractions

Seven quarters means seven pieces, each one quarter of a whole. Four of those pieces make one whole, so you can group four of them and have one whole with three quarters left over: 1 and 3/4. Nothing was added or thrown away, the same seven pieces were just grouped. That is why the two forms are always equal.

The top of a fraction being larger than the bottom is not an error, it is a signal that the amount is more than one. Reading 7/4 as a mixed number tells you how much it is at a glance, which is why recipes, measurements and answers are usually written that way. The improper form is easier to calculate with, because there is only one number to multiply or divide rather than a whole and a fraction to keep track of.

Going the other way is multiplication and addition. For 2 and 3/5, multiply the whole number by the bottom and add the top: 2 × 5 + 3 = 13, so it is 13/5. What you are really doing is rewriting the two wholes as ten fifths, then adding the three fifths you already had. Check it by turning 13/5 back: 13 ÷ 5 is 2 remainder 3, which is 2 and 3/5.

The reason this matters most is subtraction. To work out 3 and 1/4 minus 1 and 3/4 you cannot simply take 3/4 from 1/4. Rename one of the wholes in 3 and 1/4 as four quarters, so it reads 2 and 5/4, and now the bottom numbers match and the subtraction is straightforward. This renaming is the step that most often goes missing, and it is the same borrowing you already do in whole-number subtraction.